to the graph from initial and goal positions; and (d) Search over roadmap graph.
Learning Phase Query Phase
Goal Goal Node Node 7 /nit Node ¡nil Node (a) (b) (e) (d)
Another important class of sampling based algorithms is the single-query sampling-based planners. These planners attempt to solve a query as fast as possible and do not focus on the exploration of the entire Cfree. A representative for this class is the Rapidly-exploring
Random Tree (RRT). RRT was introduced in (LaVaIIe, 1998) as an efficient data
structure and sampling scheme to quickly search high-dimensional spaces that have both algebraic constraints (arising from obstacles) and differential constraints (arising from non-holonomy and dynamics). These spaces are referred as state spaces where both types
of constraints are considered. The key idea is to bias the exploration towards unexplored
portions of the space. An RRT can be intuitively considered as a Monte-Carlo way of biasing search into largest Voronoi regions. Some variations can be considered as
stochastic fractals. As with PRM, RRT is efficient in searching high-dimensional spaces
and has many versions that can suite different planning problems. The main disadvantage of RRT is that it is not able to guarantee the generation of an optimal path based on pre- defined criteria; an optimization update is required to address this point. Fortunately, for many of these algorithms, the solutions produced are not too far from optimal in practice (LaVaIIe, 2006). Brandt (2006) has made a comparison between A* and an RRT algorithm for motion planning of robots, and found that RRT is much faster than A*. Figure 2.18 shows an example of the RRT with 2000 vertex.
Other interesting lines of work include the method in (Kondo, 1991) which finds paths for 6-DoF manipulators using heuristic search techniques that limit the part of the C- space that is explored, and the planner in Ahuactzin et al. (1992) which utilizes genetic algorithms to search for a path in high-dimensional C-spaces.
Figure 2.18: Example of RRT with 2000 vertex (LaVaIIe and Kuffner, 1999)
A considerable share of the research happening in the field of motion planning focuses on
introducing more efficient extensions to RRTs. Kuffner and LaValle (2000) introduced a simple and efficient extension to the RRT algorithm called RRT-Connect for solving single-query path planning problems in high-dimensional C-spaces. The method uses a simple greedy heuristic called "Connect" to extend the tree efficiently. It works by incrementally building two RRTs rooted at the start and the goal configurations. Each of the trees explores space around itself and also advances towards its other counterpart through the use of the greedy heuristic. Based on the comparisons between RRT-Connect
and several other RRT-based variants, it is determined that for most problems, the greedy
heuristic improves the running time, often by a factor of three or four, especially for uncluttered environments. In highly cluttered environments, the Connect heuristic only slightly increases running time in comparison to not using this heuristic in constructing
the two trees. Thus, it seems that the greedy behavior is worthwhile on average. Figure 2. 1 9 shows an example of solving a simple planning problem using the RRT-Connect.
V / (a) (b) \ ?-../ ? < / (e) (d)
Figure 2.19: Growing two trees towards each other (Kuffner and LaVaIIe, 2000)
A comparison of some of these algorithms is shown in Table 2.1, which is based on the
following criteria:
1. Completeness: Complete planning approaches are guaranteed to find a solution when it exists, or correctly report failure if one does not exist (LaVaIIe, 2006). For sampling based algorithms (e.g. RRT) completeness depends on the probability to find solution, the probability of them producing solution approaches 1 as more
time is spent. Improvements to the standard RRT can be done to address this issue (Cheng and LaVaIIe, 2002). For Grid A*, finding the solution depends on the resolution of the grid that is representing the C space, low resolution grids may result in failure in finding the solution even if it exists. PRM combines both cases of being probabilistic and resolution complete, this is due to its nature of finding
the path in two phases.
2. Optimal: In addition to completeness, algorithm optimality is considered as its ability to return optimal path with respect to some metric. For single-query sampling based algorithms (e.g. RRT), they are not able to guarantee the generation of an optimal path based on pre-defined criteria; an optimization update is required to address this point. Fortunately, for many of these algorithms, the solutions produced are not too far from optimal in practice (LaVaIIe, 2006). 3. Efficient World Updates: Modifying the obstacles in the world is a very common
case. Therefore, efficiency in re-planning the path after updating the world is important. In all the algorithms reviewed in this paper, there is no perfect algorithm for this type of query. RRT is the best even though it is considered
semi-efficient in this type of queries.
4. Efficient Query Updates: In addition to world updates, query updates efficiency is important for cases like re-planning to new goals while fixing world constraints. The PRM algorithm is efficient in this type of queries, since it can reuse the roadmap that it constructed in the preprocessing phase.
5. Good DoF Scalability: The DoFs directly affect the complexity of C-spaces, thus configurations with high DoFs are not practical for solving by many algorithms.
Grid A* and visibility graph are not suitable for solving configurations with high DoFs.
6. Non-Holonomic: The capability of solving non-holonomic configurations is a key feature in path-planning algorithms, where the algorithm is not only limited of considering global constraints that are generated from explicit obstacles in the environment (Kuffher and LaVaIIe, 2000), but it is also able to address local/differential constraints that may be found in some construction equipments. Among all reviewed algorithms, RRT stands with its high abilities in solving non-
holonomic configurations.
Table 2.1: Summary of the comparison between different algorithms (Bruce, 2004)
Approach Grid A* Visibility Graph RRT Complete res yes prob Optimal grid yes no Efficient World Updates no no semi Efficient Query Updates no no semi Good DoF Scalability no no yes Non- Holonomic no no yes semi
PRM prob,res graph no yes yes
2.6.3 Re-Planning Algorithms
In the case of motion planning for equipment on construction sites, every task has a schedule, and the unknown information can be assumed to be minor or less essential to the whole plan. Therefore, a motion re-planning approach can be efficient by modifying the off-line plan based on real-time sensed data.